if tanA+tanB=P and AtanB=q express the value of cos(2A+3B) in terms of p and q.


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Question Number 33969 by mondodotto@gmail.com last updated on 28/Apr/18

$if tanA + tanB = P and AtanB = q$ $express the value of \cos (2 A + 3 B) in terms of p and q .$
	Answered by tanmay.chaudhury50@gmail.com last updated on 28/Apr/18
	$let tanA=a tanB=b cos(2A+2B+B) =cos(2A+2B).cosB−sin(2A+2B).sinB ={cos2A.cos2B−sin2A.sin2B}cosB−{sin2A. cos2B+cos2A.sin2B}.sinB put cos2A= (1−a^2 /1+a^2 ) sin2A=2a/1+a^2 simplyfy if prlblem comment pls..$
	$l e t t a n A = a t a n B = b$ $c o s (2 A + 2 B + B)$ $= c o s (2 A + 2 B) . c o s B - s i n (2 A + 2 B) . s i n B$ $= {c o s 2 A . c o s 2 B - s i n 2 A . s i n 2 B} c o s B - {s i n 2 A .$ $c o s 2 B + c o s 2 A . s i n 2 B} . s i n B$ $p u t c o s 2 A = (1 - a^{2} / 1 + a^{2}) s i n 2 A = 2 a / 1 + a^{2}$ $s i m p l y f y i f p r l b l e m c o m m e n t p l s . .$
	Commented by mondodotto@gmail.com last updated on 29/Apr/18

	$i dont undstnd$
	Commented by tanmay.chaudhury50@gmail.com last updated on 29/Apr/18

	$o k i a m s o l v i n g i n d e t a i l s a n d s h a l l u p l o a d t h e i m a g e$
	Answered by tanmay.chaudhury50@gmail.com last updated on 29/Apr/18

	$l e t t a n A = a, t a n B = b s o a + b = p, a b = q$ ${(a - b)}^{2} = {(a + b)}^{2} - 4 a b$ $a - b = \sqrt{p^{2} - 4 q} a n d a + b = p$ $s o l v i n g a = p + \sqrt{p^{2} - 4 q}$ $b = p - \sqrt{p^{2} - 4 q}$ $c o s (2 A + 3 B)$ $= c o s 2 A . c o s 3 B - s i n 2 A . s i n 3 B$ $= (\frac{1 - {t a n}^{2} A}{1 + {t a n}^{2} A}) c o s B (3 - 4 {c o s}^{2} B) - (\frac{2 t a n A}{1 + {t a n}^{2} A}) . s i n B .$ $(4 {s i n}^{2} B - 3)$ $n o w r e p l a c e t a n A b y a t a n B b y b$ $c o s B = \frac{1}{\sqrt{1 + b^{2}}} s i n B = \frac{t a n B}{s e c B} = \frac{b}{\sqrt{1 + b^{2}}}$ $\frac{1 - a^{2}}{1 + a^{2}} . \frac{1}{\sqrt{1 + b^{2}}} (3 - \frac{4}{1 + b^{2}}) - (\frac{2 a}{1 + a^{2}}) \frac{b}{\sqrt{1 + b^{2}}} . (\frac{4 b^{2}}{1 + b^{2}} - 3$ $p u t a = p + \sqrt{p^{2} - 4 q} a n d b = p - \sqrt{p^{2} 4 q}$ $s i m p l i f y$
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